Geometric relation between centrality and the impact parameter in relativistic heavy-ion collisions

We show, under general assumptions which are well satisfied in relativistic heavy-ion collisions, that the geometric relation of centrality c to the impact parameter b, namely, cπb²/σᵢₙₑₗ, holds to a very high accuracy for all but most peripheral collisions. More precisely, if $c(N)$ is the centrality of events with a multiplicity higer than N, then b is the value of the impact parameter for which the average multiplicity n ̄ \=(b) is equal to N. The corrections to this geometric formula are of the order (Δn(b)/n ̄ \=(b))², where Δn(b) is the width of the multiplicity distribution at a given value of b; hence they are very small. In other words, the centrality effectively measures the impact parameter.